Left Termination proof of /tmp/tmpCo6Wae/div.pl

Left Termination of the query pattern
div(g,g,a)
w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PrologToPiTRSViaGraphTransformerProof (⇒, 71 ms)

↳2 PiTRS

↳3 DependencyPairsProof (⇔, 51 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 AND

    ↳7 PiDP

        ↳8 UsableRulesProof (⇔, 0 ms)

        ↳9 PiDP

        ↳10 PiDPToQDPProof (⇒, 12 ms)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔, 0 ms)

        ↳13 YES

    ↳14 PiDP

        ↳15 UsableRulesProof (⇔, 0 ms)

        ↳16 PiDP

        ↳17 PiDPToQDPProof (⇔, 0 ms)

        ↳18 QDP

        ↳19 QDPSizeChangeProof (⇔, 0 ms)

        ↳20 YES

    ↳21 PiDP

        ↳22 UsableRulesProof (⇔, 0 ms)

        ↳23 PiDP

        ↳24 PiDPToQDPProof (⇒, 0 ms)

        ↳25 QDP

        ↳26 QDPOrderProof (⇔, 26 ms)

        ↳27 QDP

        ↳28 DependencyGraphProof (⇔, 0 ms)

        ↳29 TRUE

(0) Obligation:

Clauses:

div(X, s(Y), Z) :- div_s(X, Y, Z).
div_s(0, Y, 0).
div_s(s(X), Y, 0) :- lss(X, Y).
div_s(s(X), Y, s(Z)) :- ','(sub(X, Y, R), div_s(R, Y, Z)).
lss(s(X), s(Y)) :- lss(X, Y).
lss(0, s(Y)).
sub(s(X), s(Y), Z) :- sub(X, Y, Z).
sub(X, 0, X).

Query: div(g,g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: div(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: div(T1, T2, T3), SCOPE: 1, CLAUSE: 0\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="B: div_s(T7, T8, T10)\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: div_s(T7, T8, T10), SCOPE: 2, CLAUSE: 1 | div_s(T7, T8, T10), SCOPE: 2, CLAUSE: 2 | div_s(T7, T8, T10), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: div_s(T7, T8, T10), SCOPE: 2, CLAUSE: 1\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: div_s(T7, T8, T10), SCOPE: 2, CLAUSE: 2 | div_s(T7, T8, T10), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: div_s(T7, T8, T10), SCOPE: 2, CLAUSE: 2\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: div_s(T7, T8, T10), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="C: lss(T24, T25)\n\nT24 is ground\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: lss(T24, T25), SCOPE: 3, CLAUSE: 4 | lss(T24, T25), SCOPE: 3, CLAUSE: 5\n\nT24 is ground\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: lss(T24, T25), SCOPE: 3, CLAUSE: 4\n\nT24 is ground\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: lss(T24, T25), SCOPE: 3, CLAUSE: 5\n\nT24 is ground\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: lss(T36, T37)\n\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="D: ','(sub(T49, T50, X49), div_s(X49, T50, T52))\n\nT49 is ground\nT50 is ground\nX49 is free", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="E: sub(T49, T50, X49)\n\nT49 is ground\nT50 is ground\nX49 is free", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: div_s(T55, T50, T52)\n\nT50 is ground\nT55 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: sub(T49, T50, X49), SCOPE: 4, CLAUSE: 6 | sub(T49, T50, X49), SCOPE: 4, CLAUSE: 7\n\nT49 is ground\nT50 is ground\nX49 is free", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: sub(T49, T50, X49), SCOPE: 4, CLAUSE: 6\n\nT49 is ground\nT50 is ground\nX49 is free", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: sub(T49, T50, X49), SCOPE: 4, CLAUSE: 7\n\nT49 is ground\nT50 is ground\nX49 is free", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: sub(T66, T67, X77)\n\nT66 is ground\nT67 is ground\nX77 is free", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 86 [arrowhead = none ];
86 -> 3;

87 [label="EVAL with clause\ndiv(X4, s(X5), X6) :- div_s(X4, X5, X6).\nand substitutionT1 -> T7,\nX4 -> T7,\nX5 -> T8,\nT2 -> s(T8),\nT3 -> T10,\nX6 -> T10,\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 87 [arrowhead = none ];
87 -> 7;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 88 [arrowhead = none ];
88 -> 9;

89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 89 [arrowhead = none ];
89 -> 13;

90 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 90 [arrowhead = none ];
90 -> 14;

91 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 91 [arrowhead = none ];
91 -> 15;

92 [label="EVAL with clause\ndiv_s(0, X11, 0).\nand substitutionT7 -> 0,\nT8 -> T15,\nX11 -> T15,\nT10 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 92 [arrowhead = none ];
92 -> 19;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 93 [arrowhead = none ];
93 -> 20;

94 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 94 [arrowhead = none ];
94 -> 25;

95 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 95 [arrowhead = none ];
95 -> 28;

96 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 96 [arrowhead = none ];
96 -> 21;

97 [label="EVAL with clause\ndiv_s(s(X20), X21, 0) :- lss(X20, X21).\nand substitutionX20 -> T24,\nT7 -> s(T24),\nT8 -> T25,\nX21 -> T25,\nT10 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 97 [arrowhead = none ];
97 -> 32;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 98 [arrowhead = none ];
98 -> 35;

99 [label="EVAL with clause\ndiv_s(s(X46), X47, s(X48)) :- ','(sub(X46, X47, X49), div_s(X49, X47, X48)).\nand substitutionX46 -> T49,\nT7 -> s(T49),\nT8 -> T50,\nX47 -> T50,\nX48 -> T52,\nT10 -> s(T52),\nT51 -> T52", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 99 [arrowhead = none ];
99 -> 67;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 100 [arrowhead = none ];
100 -> 68;

101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 101 [arrowhead = none ];
101 -> 39;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
39 -> 102 [arrowhead = none ];
102 -> 43;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
39 -> 103 [arrowhead = none ];
103 -> 44;

104 [label="EVAL with clause\nlss(s(X32), s(X33)) :- lss(X32, X33).\nand substitutionX32 -> T36,\nT24 -> s(T36),\nX33 -> T37,\nT25 -> s(T37)", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 104 [arrowhead = none ];
104 -> 49;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 105 [arrowhead = none ];
105 -> 50;

106 [label="EVAL with clause\nlss(0, s(X38)).\nand substitutionT24 -> 0,\nX38 -> T42,\nT25 -> s(T42)", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 106 [arrowhead = none ];
106 -> 56;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 107 [arrowhead = none ];
107 -> 57;

108 [label="INSTANCE with matching:\nT24 -> T36\nT25 -> T37", fontsize=14, style = filled, fillcolor = lightgrey];
49 -> 108 [arrowhead = none , style=dashed];
108 -> 32[style=dashed];

109 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 109 [arrowhead = none ];
109 -> 58;

110 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
67 -> 110 [arrowhead = none ];
110 -> 69;

111 [label="SPLIT 2\nnew knowledge:\nT49 is ground\nT50 is ground\nT55 is ground\nreplacements:X49 -> T55", fontsize=14, style = filled, fillcolor = violet];
67 -> 111 [arrowhead = none ];
111 -> 70;

112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
69 -> 112 [arrowhead = none ];
112 -> 73;

113 [label="INSTANCE with matching:\nT7 -> T55\nT8 -> T50\nT10 -> T52", fontsize=14, style = filled, fillcolor = lightgrey];
70 -> 113 [arrowhead = none , style=dashed];
113 -> 7[style=dashed];

114 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
73 -> 114 [arrowhead = none ];
114 -> 76;

115 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
73 -> 115 [arrowhead = none ];
115 -> 77;

116 [label="EVAL with clause\nsub(s(X74), s(X75), X76) :- sub(X74, X75, X76).\nand substitutionX74 -> T66,\nT49 -> s(T66),\nX75 -> T67,\nT50 -> s(T67),\nX49 -> X77,\nX76 -> X77", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 116 [arrowhead = none ];
116 -> 79;

117 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
76 -> 117 [arrowhead = none ];
117 -> 80;

118 [label="EVAL with clause\nsub(X84, 0, X84).\nand substitutionT49 -> T72,\nX84 -> T72,\nT50 -> 0,\nX49 -> T72", fontsize=14, style = filled, fillcolor = lightblue];
77 -> 118 [arrowhead = none ];
118 -> 83;

119 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
77 -> 119 [arrowhead = none ];
119 -> 84;

120 [label="INSTANCE with matching:\nT49 -> T66\nT50 -> T67\nX49 -> X77", fontsize=14, style = filled, fillcolor = lightgrey];
79 -> 120 [arrowhead = none , style=dashed];
120 -> 69[style=dashed];

121 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
83 -> 121 [arrowhead = none ];
121 -> 85;

}

(2) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

divA_in_gga(T7, s(T8), T10) → U1_gga(T7, T8, T10, div_sB_in_gga(T7, T8, T10))
div_sB_in_gga(0, T15, 0) → div_sB_out_gga(0, T15, 0)
div_sB_in_gga(s(T24), T25, 0) → U3_gga(T24, T25, lssC_in_gg(T24, T25))
lssC_in_gg(s(T36), s(T37)) → U2_gg(T36, T37, lssC_in_gg(T36, T37))
lssC_in_gg(0, s(T42)) → lssC_out_gg(0, s(T42))
U2_gg(T36, T37, lssC_out_gg(T36, T37)) → lssC_out_gg(s(T36), s(T37))
U3_gga(T24, T25, lssC_out_gg(T24, T25)) → div_sB_out_gga(s(T24), T25, 0)
div_sB_in_gga(s(T49), T50, s(T52)) → U4_gga(T49, T50, T52, pD_in_ggaa(T49, T50, X49, T52))
pD_in_ggaa(T49, T50, T55, T52) → U6_ggaa(T49, T50, T55, T52, subE_in_gga(T49, T50, T55))
subE_in_gga(s(T66), s(T67), X77) → U5_gga(T66, T67, X77, subE_in_gga(T66, T67, X77))
subE_in_gga(T72, 0, T72) → subE_out_gga(T72, 0, T72)
U5_gga(T66, T67, X77, subE_out_gga(T66, T67, X77)) → subE_out_gga(s(T66), s(T67), X77)
U6_ggaa(T49, T50, T55, T52, subE_out_gga(T49, T50, T55)) → U7_ggaa(T49, T50, T55, T52, div_sB_in_gga(T55, T50, T52))
U7_ggaa(T49, T50, T55, T52, div_sB_out_gga(T55, T50, T52)) → pD_out_ggaa(T49, T50, T55, T52)
U4_gga(T49, T50, T52, pD_out_ggaa(T49, T50, X49, T52)) → div_sB_out_gga(s(T49), T50, s(T52))
U1_gga(T7, T8, T10, div_sB_out_gga(T7, T8, T10)) → divA_out_gga(T7, s(T8), T10)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

DIVA_IN_GGA(T7, s(T8), T10) → U1_GGA(T7, T8, T10, div_sB_in_gga(T7, T8, T10))
DIVA_IN_GGA(T7, s(T8), T10) → DIV_SB_IN_GGA(T7, T8, T10)
DIV_SB_IN_GGA(s(T24), T25, 0) → U3_GGA(T24, T25, lssC_in_gg(T24, T25))
DIV_SB_IN_GGA(s(T24), T25, 0) → LSSC_IN_GG(T24, T25)
LSSC_IN_GG(s(T36), s(T37)) → U2_GG(T36, T37, lssC_in_gg(T36, T37))
LSSC_IN_GG(s(T36), s(T37)) → LSSC_IN_GG(T36, T37)
DIV_SB_IN_GGA(s(T49), T50, s(T52)) → U4_GGA(T49, T50, T52, pD_in_ggaa(T49, T50, X49, T52))
DIV_SB_IN_GGA(s(T49), T50, s(T52)) → PD_IN_GGAA(T49, T50, X49, T52)
PD_IN_GGAA(T49, T50, T55, T52) → U6_GGAA(T49, T50, T55, T52, subE_in_gga(T49, T50, T55))
PD_IN_GGAA(T49, T50, T55, T52) → SUBE_IN_GGA(T49, T50, T55)
SUBE_IN_GGA(s(T66), s(T67), X77) → U5_GGA(T66, T67, X77, subE_in_gga(T66, T67, X77))
SUBE_IN_GGA(s(T66), s(T67), X77) → SUBE_IN_GGA(T66, T67, X77)
U6_GGAA(T49, T50, T55, T52, subE_out_gga(T49, T50, T55)) → U7_GGAA(T49, T50, T55, T52, div_sB_in_gga(T55, T50, T52))
U6_GGAA(T49, T50, T55, T52, subE_out_gga(T49, T50, T55)) → DIV_SB_IN_GGA(T55, T50, T52)

The TRS R consists of the following rules:

divA_in_gga(T7, s(T8), T10) → U1_gga(T7, T8, T10, div_sB_in_gga(T7, T8, T10))
div_sB_in_gga(0, T15, 0) → div_sB_out_gga(0, T15, 0)
div_sB_in_gga(s(T24), T25, 0) → U3_gga(T24, T25, lssC_in_gg(T24, T25))
lssC_in_gg(s(T36), s(T37)) → U2_gg(T36, T37, lssC_in_gg(T36, T37))
lssC_in_gg(0, s(T42)) → lssC_out_gg(0, s(T42))
U2_gg(T36, T37, lssC_out_gg(T36, T37)) → lssC_out_gg(s(T36), s(T37))
U3_gga(T24, T25, lssC_out_gg(T24, T25)) → div_sB_out_gga(s(T24), T25, 0)
div_sB_in_gga(s(T49), T50, s(T52)) → U4_gga(T49, T50, T52, pD_in_ggaa(T49, T50, X49, T52))
pD_in_ggaa(T49, T50, T55, T52) → U6_ggaa(T49, T50, T55, T52, subE_in_gga(T49, T50, T55))
subE_in_gga(s(T66), s(T67), X77) → U5_gga(T66, T67, X77, subE_in_gga(T66, T67, X77))
subE_in_gga(T72, 0, T72) → subE_out_gga(T72, 0, T72)
U5_gga(T66, T67, X77, subE_out_gga(T66, T67, X77)) → subE_out_gga(s(T66), s(T67), X77)
U6_ggaa(T49, T50, T55, T52, subE_out_gga(T49, T50, T55)) → U7_ggaa(T49, T50, T55, T52, div_sB_in_gga(T55, T50, T52))
U7_ggaa(T49, T50, T55, T52, div_sB_out_gga(T55, T50, T52)) → pD_out_ggaa(T49, T50, T55, T52)
U4_gga(T49, T50, T52, pD_out_ggaa(T49, T50, X49, T52)) → div_sB_out_gga(s(T49), T50, s(T52))
U1_gga(T7, T8, T10, div_sB_out_gga(T7, T8, T10)) → divA_out_gga(T7, s(T8), T10)

The argument filtering Pi contains the following mapping:
divA_in_gga(x1, x2, x3) = divA_in_gga(x1, x2)
s(x1) = s(x1)
U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4)
div_sB_in_gga(x1, x2, x3) = div_sB_in_gga(x1, x2)
0 = 0
div_sB_out_gga(x1, x2, x3) = div_sB_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3) = U3_gga(x1, x2, x3)
lssC_in_gg(x1, x2) = lssC_in_gg(x1, x2)
U2_gg(x1, x2, x3) = U2_gg(x1, x2, x3)
lssC_out_gg(x1, x2) = lssC_out_gg(x1, x2)
U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4)
pD_in_ggaa(x1, x2, x3, x4) = pD_in_ggaa(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5) = U6_ggaa(x1, x2, x5)
subE_in_gga(x1, x2, x3) = subE_in_gga(x1, x2)
U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4)
subE_out_gga(x1, x2, x3) = subE_out_gga(x1, x2, x3)
U7_ggaa(x1, x2, x3, x4, x5) = U7_ggaa(x1, x2, x3, x5)
pD_out_ggaa(x1, x2, x3, x4) = pD_out_ggaa(x1, x2, x3, x4)
divA_out_gga(x1, x2, x3) = divA_out_gga(x1, x2, x3)
DIVA_IN_GGA(x1, x2, x3) = DIVA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4)
DIV_SB_IN_GGA(x1, x2, x3) = DIV_SB_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3) = U3_GGA(x1, x2, x3)
LSSC_IN_GG(x1, x2) = LSSC_IN_GG(x1, x2)
U2_GG(x1, x2, x3) = U2_GG(x1, x2, x3)
U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4)
PD_IN_GGAA(x1, x2, x3, x4) = PD_IN_GGAA(x1, x2)
U6_GGAA(x1, x2, x3, x4, x5) = U6_GGAA(x1, x2, x5)
SUBE_IN_GGA(x1, x2, x3) = SUBE_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4)
U7_GGAA(x1, x2, x3, x4, x5) = U7_GGAA(x1, x2, x3, x5)

We have to consider all (P,R,Pi)-chains

(4) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

DIVA_IN_GGA(T7, s(T8), T10) → U1_GGA(T7, T8, T10, div_sB_in_gga(T7, T8, T10))
DIVA_IN_GGA(T7, s(T8), T10) → DIV_SB_IN_GGA(T7, T8, T10)
DIV_SB_IN_GGA(s(T24), T25, 0) → U3_GGA(T24, T25, lssC_in_gg(T24, T25))
DIV_SB_IN_GGA(s(T24), T25, 0) → LSSC_IN_GG(T24, T25)
LSSC_IN_GG(s(T36), s(T37)) → U2_GG(T36, T37, lssC_in_gg(T36, T37))
LSSC_IN_GG(s(T36), s(T37)) → LSSC_IN_GG(T36, T37)
DIV_SB_IN_GGA(s(T49), T50, s(T52)) → U4_GGA(T49, T50, T52, pD_in_ggaa(T49, T50, X49, T52))
DIV_SB_IN_GGA(s(T49), T50, s(T52)) → PD_IN_GGAA(T49, T50, X49, T52)
PD_IN_GGAA(T49, T50, T55, T52) → U6_GGAA(T49, T50, T55, T52, subE_in_gga(T49, T50, T55))
PD_IN_GGAA(T49, T50, T55, T52) → SUBE_IN_GGA(T49, T50, T55)
SUBE_IN_GGA(s(T66), s(T67), X77) → U5_GGA(T66, T67, X77, subE_in_gga(T66, T67, X77))
SUBE_IN_GGA(s(T66), s(T67), X77) → SUBE_IN_GGA(T66, T67, X77)
U6_GGAA(T49, T50, T55, T52, subE_out_gga(T49, T50, T55)) → U7_GGAA(T49, T50, T55, T52, div_sB_in_gga(T55, T50, T52))
U6_GGAA(T49, T50, T55, T52, subE_out_gga(T49, T50, T55)) → DIV_SB_IN_GGA(T55, T50, T52)

The TRS R consists of the following rules:

divA_in_gga(T7, s(T8), T10) → U1_gga(T7, T8, T10, div_sB_in_gga(T7, T8, T10))
div_sB_in_gga(0, T15, 0) → div_sB_out_gga(0, T15, 0)
div_sB_in_gga(s(T24), T25, 0) → U3_gga(T24, T25, lssC_in_gg(T24, T25))
lssC_in_gg(s(T36), s(T37)) → U2_gg(T36, T37, lssC_in_gg(T36, T37))
lssC_in_gg(0, s(T42)) → lssC_out_gg(0, s(T42))
U2_gg(T36, T37, lssC_out_gg(T36, T37)) → lssC_out_gg(s(T36), s(T37))
U3_gga(T24, T25, lssC_out_gg(T24, T25)) → div_sB_out_gga(s(T24), T25, 0)
div_sB_in_gga(s(T49), T50, s(T52)) → U4_gga(T49, T50, T52, pD_in_ggaa(T49, T50, X49, T52))
pD_in_ggaa(T49, T50, T55, T52) → U6_ggaa(T49, T50, T55, T52, subE_in_gga(T49, T50, T55))
subE_in_gga(s(T66), s(T67), X77) → U5_gga(T66, T67, X77, subE_in_gga(T66, T67, X77))
subE_in_gga(T72, 0, T72) → subE_out_gga(T72, 0, T72)
U5_gga(T66, T67, X77, subE_out_gga(T66, T67, X77)) → subE_out_gga(s(T66), s(T67), X77)
U6_ggaa(T49, T50, T55, T52, subE_out_gga(T49, T50, T55)) → U7_ggaa(T49, T50, T55, T52, div_sB_in_gga(T55, T50, T52))
U7_ggaa(T49, T50, T55, T52, div_sB_out_gga(T55, T50, T52)) → pD_out_ggaa(T49, T50, T55, T52)
U4_gga(T49, T50, T52, pD_out_ggaa(T49, T50, X49, T52)) → div_sB_out_gga(s(T49), T50, s(T52))
U1_gga(T7, T8, T10, div_sB_out_gga(T7, T8, T10)) → divA_out_gga(T7, s(T8), T10)

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 9 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

SUBE_IN_GGA(s(T66), s(T67), X77) → SUBE_IN_GGA(T66, T67, X77)

The TRS R consists of the following rules:

divA_in_gga(T7, s(T8), T10) → U1_gga(T7, T8, T10, div_sB_in_gga(T7, T8, T10))
div_sB_in_gga(0, T15, 0) → div_sB_out_gga(0, T15, 0)
div_sB_in_gga(s(T24), T25, 0) → U3_gga(T24, T25, lssC_in_gg(T24, T25))
lssC_in_gg(s(T36), s(T37)) → U2_gg(T36, T37, lssC_in_gg(T36, T37))
lssC_in_gg(0, s(T42)) → lssC_out_gg(0, s(T42))
U2_gg(T36, T37, lssC_out_gg(T36, T37)) → lssC_out_gg(s(T36), s(T37))
U3_gga(T24, T25, lssC_out_gg(T24, T25)) → div_sB_out_gga(s(T24), T25, 0)
div_sB_in_gga(s(T49), T50, s(T52)) → U4_gga(T49, T50, T52, pD_in_ggaa(T49, T50, X49, T52))
pD_in_ggaa(T49, T50, T55, T52) → U6_ggaa(T49, T50, T55, T52, subE_in_gga(T49, T50, T55))
subE_in_gga(s(T66), s(T67), X77) → U5_gga(T66, T67, X77, subE_in_gga(T66, T67, X77))
subE_in_gga(T72, 0, T72) → subE_out_gga(T72, 0, T72)
U5_gga(T66, T67, X77, subE_out_gga(T66, T67, X77)) → subE_out_gga(s(T66), s(T67), X77)
U6_ggaa(T49, T50, T55, T52, subE_out_gga(T49, T50, T55)) → U7_ggaa(T49, T50, T55, T52, div_sB_in_gga(T55, T50, T52))
U7_ggaa(T49, T50, T55, T52, div_sB_out_gga(T55, T50, T52)) → pD_out_ggaa(T49, T50, T55, T52)
U4_gga(T49, T50, T52, pD_out_ggaa(T49, T50, X49, T52)) → div_sB_out_gga(s(T49), T50, s(T52))
U1_gga(T7, T8, T10, div_sB_out_gga(T7, T8, T10)) → divA_out_gga(T7, s(T8), T10)

(8) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(9) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

SUBE_IN_GGA(s(T66), s(T67), X77) → SUBE_IN_GGA(T66, T67, X77)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
SUBE_IN_GGA(x1, x2, x3) = SUBE_IN_GGA(x1, x2)

We have to consider all (P,R,Pi)-chains

(10) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(11) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SUBE_IN_GGA(s(T66), s(T67)) → SUBE_IN_GGA(T66, T67)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

SUBE_IN_GGA(s(T66), s(T67)) → SUBE_IN_GGA(T66, T67)
The graph contains the following edges 1 > 1, 2 > 2

(13) YES

(14) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LSSC_IN_GG(s(T36), s(T37)) → LSSC_IN_GG(T36, T37)

The TRS R consists of the following rules:

divA_in_gga(T7, s(T8), T10) → U1_gga(T7, T8, T10, div_sB_in_gga(T7, T8, T10))
div_sB_in_gga(0, T15, 0) → div_sB_out_gga(0, T15, 0)
div_sB_in_gga(s(T24), T25, 0) → U3_gga(T24, T25, lssC_in_gg(T24, T25))
lssC_in_gg(s(T36), s(T37)) → U2_gg(T36, T37, lssC_in_gg(T36, T37))
lssC_in_gg(0, s(T42)) → lssC_out_gg(0, s(T42))
U2_gg(T36, T37, lssC_out_gg(T36, T37)) → lssC_out_gg(s(T36), s(T37))
U3_gga(T24, T25, lssC_out_gg(T24, T25)) → div_sB_out_gga(s(T24), T25, 0)
div_sB_in_gga(s(T49), T50, s(T52)) → U4_gga(T49, T50, T52, pD_in_ggaa(T49, T50, X49, T52))
pD_in_ggaa(T49, T50, T55, T52) → U6_ggaa(T49, T50, T55, T52, subE_in_gga(T49, T50, T55))
subE_in_gga(s(T66), s(T67), X77) → U5_gga(T66, T67, X77, subE_in_gga(T66, T67, X77))
subE_in_gga(T72, 0, T72) → subE_out_gga(T72, 0, T72)
U5_gga(T66, T67, X77, subE_out_gga(T66, T67, X77)) → subE_out_gga(s(T66), s(T67), X77)
U6_ggaa(T49, T50, T55, T52, subE_out_gga(T49, T50, T55)) → U7_ggaa(T49, T50, T55, T52, div_sB_in_gga(T55, T50, T52))
U7_ggaa(T49, T50, T55, T52, div_sB_out_gga(T55, T50, T52)) → pD_out_ggaa(T49, T50, T55, T52)
U4_gga(T49, T50, T52, pD_out_ggaa(T49, T50, X49, T52)) → div_sB_out_gga(s(T49), T50, s(T52))
U1_gga(T7, T8, T10, div_sB_out_gga(T7, T8, T10)) → divA_out_gga(T7, s(T8), T10)

(15) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(16) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

LSSC_IN_GG(s(T36), s(T37)) → LSSC_IN_GG(T36, T37)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains

(17) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(18) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LSSC_IN_GG(s(T36), s(T37)) → LSSC_IN_GG(T36, T37)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(19) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

LSSC_IN_GG(s(T36), s(T37)) → LSSC_IN_GG(T36, T37)
The graph contains the following edges 1 > 1, 2 > 2

(20) YES

(21) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

DIV_SB_IN_GGA(s(T49), T50, s(T52)) → PD_IN_GGAA(T49, T50, X49, T52)
PD_IN_GGAA(T49, T50, T55, T52) → U6_GGAA(T49, T50, T55, T52, subE_in_gga(T49, T50, T55))
U6_GGAA(T49, T50, T55, T52, subE_out_gga(T49, T50, T55)) → DIV_SB_IN_GGA(T55, T50, T52)

The TRS R consists of the following rules:

divA_in_gga(T7, s(T8), T10) → U1_gga(T7, T8, T10, div_sB_in_gga(T7, T8, T10))
div_sB_in_gga(0, T15, 0) → div_sB_out_gga(0, T15, 0)
div_sB_in_gga(s(T24), T25, 0) → U3_gga(T24, T25, lssC_in_gg(T24, T25))
lssC_in_gg(s(T36), s(T37)) → U2_gg(T36, T37, lssC_in_gg(T36, T37))
lssC_in_gg(0, s(T42)) → lssC_out_gg(0, s(T42))
U2_gg(T36, T37, lssC_out_gg(T36, T37)) → lssC_out_gg(s(T36), s(T37))
U3_gga(T24, T25, lssC_out_gg(T24, T25)) → div_sB_out_gga(s(T24), T25, 0)
div_sB_in_gga(s(T49), T50, s(T52)) → U4_gga(T49, T50, T52, pD_in_ggaa(T49, T50, X49, T52))
pD_in_ggaa(T49, T50, T55, T52) → U6_ggaa(T49, T50, T55, T52, subE_in_gga(T49, T50, T55))
subE_in_gga(s(T66), s(T67), X77) → U5_gga(T66, T67, X77, subE_in_gga(T66, T67, X77))
subE_in_gga(T72, 0, T72) → subE_out_gga(T72, 0, T72)
U5_gga(T66, T67, X77, subE_out_gga(T66, T67, X77)) → subE_out_gga(s(T66), s(T67), X77)
U6_ggaa(T49, T50, T55, T52, subE_out_gga(T49, T50, T55)) → U7_ggaa(T49, T50, T55, T52, div_sB_in_gga(T55, T50, T52))
U7_ggaa(T49, T50, T55, T52, div_sB_out_gga(T55, T50, T52)) → pD_out_ggaa(T49, T50, T55, T52)
U4_gga(T49, T50, T52, pD_out_ggaa(T49, T50, X49, T52)) → div_sB_out_gga(s(T49), T50, s(T52))
U1_gga(T7, T8, T10, div_sB_out_gga(T7, T8, T10)) → divA_out_gga(T7, s(T8), T10)

The argument filtering Pi contains the following mapping:
divA_in_gga(x1, x2, x3) = divA_in_gga(x1, x2)
s(x1) = s(x1)
U1_gga(x1, x2, x3, x4) = U1_gga(x1, x2, x4)
div_sB_in_gga(x1, x2, x3) = div_sB_in_gga(x1, x2)
0 = 0
div_sB_out_gga(x1, x2, x3) = div_sB_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3) = U3_gga(x1, x2, x3)
lssC_in_gg(x1, x2) = lssC_in_gg(x1, x2)
U2_gg(x1, x2, x3) = U2_gg(x1, x2, x3)
lssC_out_gg(x1, x2) = lssC_out_gg(x1, x2)
U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4)
pD_in_ggaa(x1, x2, x3, x4) = pD_in_ggaa(x1, x2)
U6_ggaa(x1, x2, x3, x4, x5) = U6_ggaa(x1, x2, x5)
subE_in_gga(x1, x2, x3) = subE_in_gga(x1, x2)
U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4)
subE_out_gga(x1, x2, x3) = subE_out_gga(x1, x2, x3)
U7_ggaa(x1, x2, x3, x4, x5) = U7_ggaa(x1, x2, x3, x5)
pD_out_ggaa(x1, x2, x3, x4) = pD_out_ggaa(x1, x2, x3, x4)
divA_out_gga(x1, x2, x3) = divA_out_gga(x1, x2, x3)
DIV_SB_IN_GGA(x1, x2, x3) = DIV_SB_IN_GGA(x1, x2)
PD_IN_GGAA(x1, x2, x3, x4) = PD_IN_GGAA(x1, x2)
U6_GGAA(x1, x2, x3, x4, x5) = U6_GGAA(x1, x2, x5)

We have to consider all (P,R,Pi)-chains

(22) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(23) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

DIV_SB_IN_GGA(s(T49), T50, s(T52)) → PD_IN_GGAA(T49, T50, X49, T52)
PD_IN_GGAA(T49, T50, T55, T52) → U6_GGAA(T49, T50, T55, T52, subE_in_gga(T49, T50, T55))
U6_GGAA(T49, T50, T55, T52, subE_out_gga(T49, T50, T55)) → DIV_SB_IN_GGA(T55, T50, T52)

The TRS R consists of the following rules:

subE_in_gga(s(T66), s(T67), X77) → U5_gga(T66, T67, X77, subE_in_gga(T66, T67, X77))
subE_in_gga(T72, 0, T72) → subE_out_gga(T72, 0, T72)
U5_gga(T66, T67, X77, subE_out_gga(T66, T67, X77)) → subE_out_gga(s(T66), s(T67), X77)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
0 = 0
subE_in_gga(x1, x2, x3) = subE_in_gga(x1, x2)
U5_gga(x1, x2, x3, x4) = U5_gga(x1, x2, x4)
subE_out_gga(x1, x2, x3) = subE_out_gga(x1, x2, x3)
DIV_SB_IN_GGA(x1, x2, x3) = DIV_SB_IN_GGA(x1, x2)
PD_IN_GGAA(x1, x2, x3, x4) = PD_IN_GGAA(x1, x2)
U6_GGAA(x1, x2, x3, x4, x5) = U6_GGAA(x1, x2, x5)

We have to consider all (P,R,Pi)-chains

(24) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(25) Obligation:

Q DP problem:
The TRS P consists of the following rules:

DIV_SB_IN_GGA(s(T49), T50) → PD_IN_GGAA(T49, T50)
PD_IN_GGAA(T49, T50) → U6_GGAA(T49, T50, subE_in_gga(T49, T50))
U6_GGAA(T49, T50, subE_out_gga(T49, T50, T55)) → DIV_SB_IN_GGA(T55, T50)

The TRS R consists of the following rules:

subE_in_gga(s(T66), s(T67)) → U5_gga(T66, T67, subE_in_gga(T66, T67))
subE_in_gga(T72, 0) → subE_out_gga(T72, 0, T72)
U5_gga(T66, T67, subE_out_gga(T66, T67, X77)) → subE_out_gga(s(T66), s(T67), X77)

The set Q consists of the following terms:

subE_in_gga(x0, x1)
U5_gga(x0, x1, x2)

We have to consider all (P,Q,R)-chains.

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

PD_IN_GGAA(T49, T50) → U6_GGAA(T49, T50, subE_in_gga(T49, T50))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(DIV_SB_IN_GGA(x₁, x₂)) = x₁ + x₂
POL(PD_IN_GGAA(x₁, x₂)) = 1 + x₁ + x₂
POL(U5_gga(x₁, x₂, x₃)) = 1 + x₃
POL(U6_GGAA(x₁, x₂, x₃)) = x₃
POL(s(x₁)) = 1 + x₁
POL(subE_in_gga(x₁, x₂)) = x₁
POL(subE_out_gga(x₁, x₂, x₃)) = x₂ + x₃

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

subE_in_gga(s(T66), s(T67)) → U5_gga(T66, T67, subE_in_gga(T66, T67))
subE_in_gga(T72, 0) → subE_out_gga(T72, 0, T72)
U5_gga(T66, T67, subE_out_gga(T66, T67, X77)) → subE_out_gga(s(T66), s(T67), X77)

(27) Obligation:

Q DP problem:
The TRS P consists of the following rules:

DIV_SB_IN_GGA(s(T49), T50) → PD_IN_GGAA(T49, T50)
U6_GGAA(T49, T50, subE_out_gga(T49, T50, T55)) → DIV_SB_IN_GGA(T55, T50)

The TRS R consists of the following rules:

subE_in_gga(s(T66), s(T67)) → U5_gga(T66, T67, subE_in_gga(T66, T67))
subE_in_gga(T72, 0) → subE_out_gga(T72, 0, T72)
U5_gga(T66, T67, subE_out_gga(T66, T67, X77)) → subE_out_gga(s(T66), s(T67), X77)

The set Q consists of the following terms:

subE_in_gga(x0, x1)
U5_gga(x0, x1, x2)

We have to consider all (P,Q,R)-chains.

(28) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(0) Obligation:

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

(2) Obligation:

(3) DependencyPairsProof (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) PiDPToQDPProof (SOUND transformation)

(11) Obligation:

(12) QDPSizeChangeProof (EQUIVALENT transformation)

(13) YES

(14) Obligation:

(15) UsableRulesProof (EQUIVALENT transformation)

(16) Obligation:

(17) PiDPToQDPProof (EQUIVALENT transformation)

(18) Obligation:

(19) QDPSizeChangeProof (EQUIVALENT transformation)

(20) YES

(21) Obligation:

(22) UsableRulesProof (EQUIVALENT transformation)

(23) Obligation:

(24) PiDPToQDPProof (SOUND transformation)

(25) Obligation:

(26) QDPOrderProof (EQUIVALENT transformation)

(27) Obligation:

(28) DependencyGraphProof (EQUIVALENT transformation)

(29) TRUE